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In negative contexts, “much” and “many” denote a small amount or number of uncountable and countable nouns respectively. “Any” implies the absence of something, applicable to both countable and uncountable nouns. For instance, “Not many companies can build planes” implies a small number of such companies. “There’s not much difference between them” suggests a minimal difference. “I don’t have any apples” means zero apples are present.